In December 1996, Bernhard Schölkopf, Alexander Smola, and Klaus-Robert Müller introduced a method for nonlinear principal component analysis (PCA) using kernel functions. This approach allows for the efficient computation of principal components in high-dimensional feature spaces, which are related to input space via a nonlinear mapping. The method leverages kernel functions to express dot products in feature space, enabling the construction of nonlinear versions of algorithms that rely solely on dot products. The paper discusses the derivation of this method, its application to nonlinear feature extraction for pattern recognition, and compares it with other techniques. It also presents experimental results on nonlinear PCA, demonstrating its effectiveness in tasks such as object recognition and character recognition. The authors highlight the advantages of kernel PCA over traditional PCA, including the ability to extract more principal components and the use of nonlinear kernels to capture higher-order statistics. The paper also touches on applications in clustering, classification, and image indexing, emphasizing the flexibility and efficiency of kernel-based methods. The authors conclude that kernel PCA provides a powerful tool for nonlinear feature extraction and classification, with potential for further improvements through the integration of additional techniques.In December 1996, Bernhard Schölkopf, Alexander Smola, and Klaus-Robert Müller introduced a method for nonlinear principal component analysis (PCA) using kernel functions. This approach allows for the efficient computation of principal components in high-dimensional feature spaces, which are related to input space via a nonlinear mapping. The method leverages kernel functions to express dot products in feature space, enabling the construction of nonlinear versions of algorithms that rely solely on dot products. The paper discusses the derivation of this method, its application to nonlinear feature extraction for pattern recognition, and compares it with other techniques. It also presents experimental results on nonlinear PCA, demonstrating its effectiveness in tasks such as object recognition and character recognition. The authors highlight the advantages of kernel PCA over traditional PCA, including the ability to extract more principal components and the use of nonlinear kernels to capture higher-order statistics. The paper also touches on applications in clustering, classification, and image indexing, emphasizing the flexibility and efficiency of kernel-based methods. The authors conclude that kernel PCA provides a powerful tool for nonlinear feature extraction and classification, with potential for further improvements through the integration of additional techniques.